NEW YORK, June 29 (Reuters) - The U.S. Federal Reserve is starting to unwind a multi-trillion-dollar program that has propped up bond markets for nearly a decade. But ask bond traders what the impact will be, or how they are strategizing around the so-called "taper," and get little more than a yawn.

In interviews, trading executives at Wall Street banks said they are not expecting much to happen just because the Fed is finally whittling down its balance sheet. Investors largely expected the move, and want to see other policy changes from Washington and from central banks abroad before shifting their portfolios, they said.

"Because the Fed has been talking about the taper for four years, people have heard enough about it," Chris Leonard, head of U.S. rates trading at Barclays PLC, told Reuters. "They feel like they're prepared."

On June 14, the Federal Reserve raised interest rates for the third time since December, and said it would start reducing its $4.2 trillion portfolio of Treasury bonds, mortgage-backed securities and federal agency debt that it had purchased to support markets and the broader economy.

This week, central bankers in Europe indicated that they may start tightening monetary policy as well, offering a quick shot in the arm for bond markets.

But the process is expected to take a long time, trading executives said. The majority of assets the Fed plans to wind down are not scheduled to mature for at least five years. Europe has only begun to talk about hiking rates and being less accommodative.

Instead of cheering the moves, trading executives pointed to signs that the bond market does not much care about the Fed's balance sheet shrinkage, and noted that upticks in bond yields like those following comments from central bankers in England and Europe this week have not lasted long in recent years.

Volatility remains near historic lows, as do 10-year Treasury bond yields. And the difference between short-term and long-term interest rates has narrowed, suggesting there are concerns about the long-term health of the economy. tmsnrt.rs/2tu88A0

All of this bodes poorly for second-quarter profits at big Wall Street banks like JPMorgan Chase & Co, Citigroup Inc, Bank of America Corp, Goldman Sachs Group Inc and Morgan Stanley, whose bond trading businesses have been suffering for years due to a combination of low rates, weak trading activity and new regulations.

At recent industry conferences, senior bank executives have told analysts and investors to expect declines of 10 to 15 percent in trading revenue when they report second-quarter results next month, due in large part to weak bond market activity. Stock analysts have been issuing gloomier forecasts for similar reasons.

"Going into the slower part of the year, we still see some value, but the group has its work cut out for it to justify '18 estimates," Evercore ISI analyst Glenn Schorr wrote about the Wall Street banks he covers in a Wednesday report cutting profit forecasts.

The latest troubles come on top of a 51 percent decline in annual bond trading revenue across Wall Street since 2009, according to research firm Coalition.

For conditions to change, investors need to see more evidence that the U.S. economy is poised for meaningful growth and that other global economies are truly on a stronger footing, trading executives said.

One head of U.S. Treasury trading said his clients were more disappointed that the White House and Congress had not been able to make progress on tax reform or infrastructure spending than they were encouraged by the Fed's widely expected decision.

In the meantime, there is skepticism that economic conditions will improve as much and as quickly as central banks have been indicating, said the executive, who was not authorized to speak publicly.

"The market was betting on (growth) heavily going into 2017 and in the meantime oil has declined and inflation has declined," he said. "It's almost as of the market wants to see it before it prices it in."